Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 58363 8 pages doi 2007 58363 Research Article Superlinear Equations Involving Nonlinearities Limited by Asymptotically Homogeneous Functions Sebastian Lorca Marco Aurelio Souto and Pedro Ubilla Received 24 August 2006 Revised 24 November 2006 Accepted 28 March 2007 Recommended by Y. Giga We obtain a solution of the quasilinear equation -Apu f u in o u 0 on do. Here the nonlinearity f is superlinear at zero and it is located near infinity between two functions that belong to a class of functions where the Ambrosetti-Rabinowitz condition is satisfied. More precisely we consider the class of functions that are asymptotically homogeneous of index q. Copyright 2007 Sebastian Lorca et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Consider the problem -Apu f u in o Zan u 0 on do. Here o is a bounded smooth domain in RN with N 3 and 1 p N. We assume that f R R is a locally Lipschitz function satisfying the condition fl lim. _ f s sp-1 0. It is well known that problems involving the p-Laplacian operator appear in many contexts. Some of these problems come from different areas of applied mathematics and physics. For example they may be found in the study of non-Newtonian fluids nonlinear elasticity and reaction diffusions. For discussions about problems modelled by these boundary value problems see for example 1 . One of the most widely used results for solving problem is the mountain pass theorem. In order to apply this theorem it is necessary that the Euler-Lagrange functional associated to the problem has the Palais-Smale property. One way to ensure this is to 2 Journal of Inequalities and Applications assume that f satisfies some Ambrosetti-Rabinowitz-type condition see . 2 or 3
